Georgia Institute of Technology College of Sciences

Given n points on a disk, we will describe how to build an A-infinity category based on the instanton Floer complex of links, and explain why it is finitely generated. This is based on work in progress with Ko Honda.

We study a particular distinguished component (the 'Hitchin component') of the space of surface group representations to SL(3,\R).  In this setting, both Hitchin (via Higgs bundles) and the more ancient subject of affine spheres associate a bundle of holomorphic differentials over Teichmuller space to this component of the character variety.  We focus on a ray of holomorphic differentials and provide a formula, tropical in appearance, for the asymptotic holonomy of the representations in terms of the local geometry of the differential.  Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. All of this is joint work with John Loftin and Andrea Tamburelli, and all the constructions and definitions will be (likely briskly) explained.

After an introduction to how to think about the mapping class groupand its cohomology, I will discuss a recent theorem of mine saying
that passing to the level-l subgroup does not change the rational cohomology in a stable range.

Two of the most well-studied topics in geometric group theory are CAT(0) cube complexes and mapping class groups. This is in part because they both admit powerful combinatorial-like structures that encode their (coarse) geometry: hyperplanes for the former and curve graphs for the latter. In recent years, analogies between the two theories have become more apparent. For instance: there are counterparts of curve graphs for CAT(0) cube complexes and rigidity theorems for such counterparts that mirror the surface setting, and both can be studied using the machinery of hierarchical hyperbolicity. However, the considerably larger class of CAT(0) spaces is left out of this analogy, as the lack of a combinatorial-like structure presents a difficulty in importing techniques from those areas. In this talk, I will speak about recent work with Petyt and Spriano where we bring CAT(0) spaces into the picture by developing analogues of hyperplanes and curve graphs for them. The talk will be accessible to everyone, and all the aforementioned terms will be defined.